(sorry it's sideways, I couldn't get it to flip)
We talked in class when we were trying to make a definition for "half" about how it isn't always immediately evident to students that the parts have to all be the same when cutting a whole into fractions. This week, during the daily Big Four, my MT once again included a problem about fractions, as she does every week, and I noticed one of my students had drawn his representations seen in the picture. I find it interesting that when he ran into the issue of cutting the circle into 1/5s, he didn't erase the circle to make a bar, but rather cut one of the 1/4s he had already made in half to represent his five parts. This leads me to believe that even though he has seen others draw fractions by using the bar method, he is not yet thinking of fractions outside of the initial explanation of "pieces of the pie" presented by my mentor teacher. I feel as though this means that although the student understands that fractions mean that the whole is cut into the denominator's number of pieces, he does not yet understand the importance of the pieces being equal in size for the fraction to truly be correct. I think if he were to learn the bar method, it would be easier for him to see where to cut it to make the five pieces. However, it could also be a result of the fact that five is an odd number, and it is more difficult to visualize where to cut odd numbers, since you can't do halves, then halves again. I think I would need to see more representations of odd-denominator fractions from this student to know for sure. The questions I would ask are:
1. Does he know that the pieces are supposed to be equal, and as close to equal as possible in drawings?
2. Does the student understand and know how to use the bar method of fraction representation?
3. What is a good way to teach explicitly that fractions need to be in equal pieces?
For questions 1 and 2, I would need to either further observe the students work, or have a conversation with the student himself or my mentor teacher about what he understands. This could either be an observation of some specific fraction work, like I described above, or just a conversation about a picture of fractions to refer to. For question three, I could get suggestions from the internet, as well as from the readings we have done this semester about fractions. I think it is something that we need to teach, but as we saw in class, it is somewhat difficult to explain to students so that they really understand what you mean. It will probably come with experience, as well as any research I might be able to do into articles and internet suggestions.
This is a perfect instantiation of some of the things we have talked about in class...I'm glad you were able to capture and post this example.
ReplyDeleteIn terms of question 3, remember that the big theme for TE 402 is not so much that there is "one right way" to teach fractions (or any one concept) and the job of TE 402 is to teach you the "right way". Rather, the emphasis is on the importance of empowering students to explore the mathematical concepts (rather than simply telling them about the concepts). So, I think the way to help this student develop his understanding is to give him more opportunities to think about fractions in different ways / different representations.